Bert Fristedt

The structure of random partitions of large integers

Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches ∞. The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct

Intersections and limits of regenerative sets

SummaryRegenerative subsets of ℝ constitute an analog of classical renewal processes. Limits and intersections of independent regenerative sets are discussed. These ideas are related to the usual quantities associated with subordinators.

The set of real numbers left uncovered by random covering intervals

SummaryRandom covering intervals are placed on the real line in a Poisson manner. Lebesgue measure governs their (random) locations and an arbitrary measure μ governs their (random) lengths. The uncovered set is a regenerative set in the sense of Hoffmann-Jørgensens generalization of regenerative phenomena introduced by Kingman. Thus, as has previously been obtained by Mandelbrot, it is the closure of the image of a subordinator —one that is identified explicitly. Well-known facts about subordinators give Shepps necessary and sufficient condition on μ for complete coverage and, when the coverage is not complete, a formula for the Hausdorff dimension of the uncovered set. The method does not seem to be applicable when the covering is not done in a Poisson manner or if the line is replaced by the plane or higher dimensional space.

The packing measure of a general subordinator

SummaryPrecise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinatorX(t) is equivalent to the upper local growth problem forY(t)=min (Y1(t), Y2(t)), whereY1 andY2 are independent copies ofX. A finite and positive packing measure is possible for subordinators “close to Cauchy”; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth ofY (although, as is well known, there is no such function for the subordinatorX itself).

SEARCHING FOR A PARTICLE ON THE REAL LINE

In this paper we consider two optimization problems and two game problems. In each problem, a particle is hidden on the real line (sometimes randomly, and sometimes by an antagonistic hider), and a seeker, starting at the origin, wishes to find the particle with minimal expected cost. We consider a fairly wide class of cost functions depending upon the position of the particle and the time used to discover it. For the games we obtain the values and (e-) optimal strategies. For the optimization problems we obtain qualitative features of (e-) optimal searches.

Constructions of local time for a Markov process

SummaryA sequence of functions defined on the space of excursions of a Markov process from a fixed point is considered. For each of the functions the sum over the excursions that begin by time t is normalized in an appropriate manner. Conditions are obtained for the convergence of the sequence of normalized sums to the local time evaluated at time t. We obtain a unified structure for convergence theorems which includes some new constructions of local time as well as many constructions previously obtained by quite varied techniques.

THE NUMBER OF EXTREME POINTS IN THE CONVEX HULL OF A RANDOM SAMPLE

Let (Xk} be an i.i.d. sequence taking values in R2 with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of (XI, . *, X } is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.

Filtering and Prediction: A Primer

Preliminaries Markov chains Filtering of discrete Markov chains Conditional expectations Filtering of continuous-space Markov chains Wiener process and continuous time filtering Stationary sequences Prediction of stationary sequences Bibliography Index.

Variations of processes with stationary, independent increments

Abstract : Variational sums for arbitrary processes with stationary, independent increments are shown to converge in probability to certain other such processes as the mesh of a sequence of partitions tend to zero. Under less restrictive conditions, centered variational sums are shown to converge. In cases of divergence, rates of divergence are found. (Author)

Hide and seek in a subset of the real line

This paper is concerned with a zero-sum two-person game in which one player, the hider, hides a particle in a given subset, known to both players, of the real line. The seeker, starting at the origin and travelling with speed no larger than one, tries to arrive at the particle as soon as possible. The payoff is the time the seeker requires to arrive at the particle divided by the distance of the particle from the origin.Analytic characterizations are obtained for the value and optimal strategies of the two players. Explicit solutions are obtained for some examples.